C₅×#7422 #11930142

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7422

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1347
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂×M₃₂	C₂²×C₅
prime radical	C₅×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

150 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,7)(8,18,21,12)(9,23)(10,26)(11,17)(13,19,31,34)(14,24)(15,20)(16,27)(22,30,37,33)(25,32)(28,35,29,36);(7,12)(8,19)(9,20)(10,27)(13,15)(14,31)(16,33)(17,18)(21,34)(22,36)(23,29)(24,25)(26,30)(28,32)(35,37)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 410);